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Single Idea 10598
[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness
]
Full Idea
A theory is 'negation complete' if it decides every sentence of its language (either the sentence, or its negation).
Gist of Idea
A theory is 'negation complete' if it proves all sentences or their negation
Source
Peter Smith (Intro to Gödel's Theorems [2007], 03.4)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.24
The
14 ideas
with the same theme
[all the truths of a system are formally deducible]:
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9544
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A system is 'weakly' complete if all wffs are derivable, and 'strongly' if theses are maximised
[Hughes/Cresswell]
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19065
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Soundness and completeness proofs test the theory of meaning, rather than the logic theory
[Dummett]
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9720
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A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
[Enderton]
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10763
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Completeness and compactness together give axiomatizability
[Tharp]
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10834
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Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
[Boolos]
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10069
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A theory is 'negation complete' if one of its sentences or its negation can always be proved
[Smith,P]
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10598
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A theory is 'negation complete' if it proves all sentences or their negation
[Smith,P]
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10597
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'Complete' applies both to whole logics, and to theories within them
[Smith,P]
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13628
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We can live well without completeness in logic
[Shapiro]
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13698
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In a complete logic you can avoid axiomatic proofs, by using models to show consequences
[Sider]
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10127
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A 'complete' theory contains either any sentence or its negation
[George/Velleman]
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10161
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If a sentence holds in every model of a theory, then it is logically derivable from the theory
[Feferman/Feferman]
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13538
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If a theory is complete, only a more powerful language can strengthen it
[Wolf,RS]
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10761
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Completeness can always be achieved by cunning model-design
[Rossberg]
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